Discrete schemes for processes in random media

We present discrete schemes for processes in random media. We prove two results. The first one is the convergence of Sinai's random walks in random environments to the Brox model. The second one is the convergence of random walks in media with random “gates” to a continuous process in a Poisson potential. The proofs are based on the following idea: we consider the discrete media as random potentials for continuous models. Received: 6 May 1999 / Revised version: 18 October 1999 / Published online: 20 October 2000     

Gumbel fluctuations for cover times in the discrete torus

This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in Aldous and Fill (Reversible Markov chains and random walks on graphs, in preparation). We also derive some corollaries which qualitatively describe “how” covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman (Ann Math (2) 171(3):2039–2087, 2010), and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest.     

Weak convergence to fractional Brownian motion in Brownian scenery

 Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries. In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery. Received: 17 April 2002 / Revised version: 11 October 2002 / Published online: 15 April 2003 Research supported by NSFC (10131040). Mathematics Subject Classification (2002): 60J55, 60J15, 60J65 Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery     

圈上的多重懒惰随机游走

本文考虑了n个定点的圈上的多重懒惰随机游走.利用偶和方法证明了其最大相遇时的期望的阶数为h_(max)×log n,其中h_(max)为圈上的一简单随机游走的最大击中时.     

Dynamic Random Walks on Heisenberg Groups

We prove a Guivarc’h law of large numbers and a central limit theorem for dynamic random walks on Heisenberg groups. The limiting distribution is explicitely given. To our knowledge this is the first study of dynamic random walks on non-commutative Lie groups.      

A note on the Poisson boundary of lamplighter random walks

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups Γ endowed with a “rich” boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich (Ann. Math. 152:659–692, 2000). A geometrical method for constructing the strip as a subset of the lamplighter group \mathbb Z₂\wr G{\mathbb {Z}_{2}\wr \Gamma} starting with a “smaller” strip in the group Γ is developed. Then, this method is applied to several classes of base groups Γ: groups with infinitely many ends, hyperbolic groups in the sense of Gromov, and Euclidean lattices. We show that under suitable hypothesis the Poisson boundary for a class of random walks on lamplighter groups is the space of infinite limit configurations.     

Slowdown estimates and central limit theorem for random walks in random environment

This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on ℤ ^d , when d≥2. We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important role in the investigation of both problems. This article also improves the previous work of the author [24], concerning estimates of probabilities of slowdowns for walks which are neutral or biased to the right. Received May 31, 1999 / final version received January 18, 2000?Published online April 19, 2000     

Limit theorems for reinforced random walks on certain trees

Consider a linearly edge-reinforced random walk defined on the b-ary tree, b≥70. We prove the strong law of large numbers for the distance of this process from the root. We give a sufficient condition for this strong law to hold for general edge-reinforced random walks and random walks in a random environment. We also provide a central limit theorem. Supported in part by a Purdue Research Foundation fellowship this work is part of the author's PhD thesis.     

On estimating the derivatives of symmetric diffusions in stationary random environment,with applications to ∇<Emphasis Type="Italic">ϕ</Emphasis> interface model

We consider diffusions on ℝ^d or random walks on ℤ^d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients. We show existence and H?lder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives. In the case of random walks, these estimates are applied to the Ginzburg-Landau ∇ϕ interface model.     

Isomorphism of random walks

We show that any two aperiodic, recurrent random walks on the integers whose jump distributions have finite seventh moment, are isomorphic as infinite measure preserving transformations. The method of proof involved uses a notion of equivalence of renewal sequences, and the “relative” isomorphism of Bernoulli shifts respecting a common state lumping with the same conditional entropy. We also prove an analogous result for random walks on the two dimensional integer lattice.     

The spectra of lamplighter groups and Cayley machines

We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr , with G a finite group, by calculating the Kesten—von Neumann—Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and Żuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by an automaton coincides with the Kesten—von Neumann—Serre spectral measure.     

Branching Random Walks in Space–Time Random Environment: Survival Probability,Global and Local Growth Rates

We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the d-dimensional integer lattice, while at each time unit, they split into independent copies according to time–space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rate is given by the directional free energy.     

ON THE CONVERGENCE OF GODUNOV SCHEME FOR NONLINEAR HYPERBOLIC SYSTEMS

51. IntroductionWe consider the one dimensional Cauchy problem for an n x n system of the formHere A(u) is a smooth matrix valued map from a domain U C R" into R", and (x, t) eR x R . The system is assumed to be strictly hyperbolic, i.e. the matrix A(u) has n realand strictly different eigenvalues at each point u C U.We note that even for smooth data a classical solution is only defined locally in time.In general the solution will develop discontinuities in finite time and it is not cle…     

A combinatorial result with applications to self-interacting random walks

We give a series of combinatorial results that can be obtained from any two collections (both indexed by Z×N) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-excited random walks and for random walks in random environments in all dimensions.     

Entropy and<Emphasis Type="Italic">σ</Emphasis>-algebra equivalence of certain random walks on random sceneries

LetX=X ₀,X ₁,…be a stationary sequence of random variables defining a sequence space Σ with shift mapσ and let (T _t, Ω) be an ergodic flow. Then the endomorphismT _X(x, ω)=(σ(x),T _{x 0}(ω)) is known as a random walk on a random scenery. In [4], Heicklen, Hoffman and Rudolph proved that within the class of random walks on random sceneries whereX is an i.i.d. sequence of Bernoulli-(1/2, 1/2) random variables, the entropy ofT _t is an isomorphism invariant. This paper extends this result to a more general class of random walks, which proves the existence of an uncountable family of smooth maps on a single manifold, no two of which are measurably isomorphic. This research was sustained in part by fellowship support from the National Physical Science Consortium and the National Security Agency.     

Large deviations for random walks in a mixing random environment and other (non‐Markov) random walks

We extend a recent work by S. R. S. Varadhan [8] on large deviations for random walks in a product random environment to include more general random walks on the lattice. In particular, some reinforced random walks and several classes of random walks in Gibbs fields are included. © 2004 Wiley Periodicals, Inc.     

The monotonicity of the probability of return into the source in models of branching random walks

The paper discusses two models of a branching random walk on a many-dimensional lattice with birth and death of particles at a single node being the source of branching. The random walk in the first model is assumed to be symmetric. In the second model an additional parameter is introduced which enables “artificial” intensification of the prevalence of branching or walk at the source and, as the result, violating the symmetry of the random walk. The monotonicity of the return probability into the source is proved for the second model, which is a key property in the analysis of branching random walks.     

Current Fluctuations for Independent Random Walks in Multiple Dimensions

Consider a system of particles evolving as independent and identically distributed (i.i.d.) random walks. Initial fluctuations in the particle density get translated over time with velocity [(v)\vec]\vec{v}, the common mean velocity of the random walks. Consider a box centered around an observer who starts at the origin and moves with constant velocity [(v)\vec]\vec{v}. To observe interesting fluctuations beyond the translation of initial density fluctuations, we measure the net flux of particles over time into this moving box. We call this the “box-current” process.     

Large deviation principles for random walks with regularly varying distributions of jumps

We establish the local and so-called “extended” large deviation principles (see [1, 2]) for random walks whose jumps fail to satisfy Cramér’s condition but have distributions varying regularly at infinity.